Analytic torsion of nilmanifolds with (2, 3, 5) distributions

1 Introduction

The classical Ray-Singer analytic torsion [45] is a spectral invariant extracted from the de Rham complex of a closed manifold. The celebrated Cheeger-Müller theorem [7,17,18,42] asserts that this analytic torsion essentially coincides with the Reidemeister torsion, a topological invariant.

Rumin and Seshadri [54] have introduced an analytic torsion of the Rumin complex on contact manifolds [48,49,51] and showed that it coincides with the Ray-Singer torsion for three-dimensional CR Seifert manifolds. Further computations for contact spheres and lens spaces have been carried out by Kitaoka [36,37]. Recently, Albin and Quan [1] proved that the Rumin-Seshadri analytic torsion differs from the Ray-Singer torsion by the integral of a local quantity, which has yet to be identified explicitly.

An analytic torsion for the Rumin complex on more general filtered manifolds [27,50,52,53] has been proposed in [29]. This analytic torsion is only defined if the osculating algebras of the filtered manifold have pure cohomology, i.e., if the grading automorphism acts by a scalar on the Lie algebra cohomology in each degree. The latter assumption appears to be rather restrictive [29, Section 3.7]. We are only aware of three types of filtered manifolds with this property: trivially filtered manifolds giving rise to the Ray-Singer torsion, contact manifolds giving rise the Rumin-Seshadri torsion, and generic rank two distributions in dimension five which are also known as (2, 3, 5) distributions [29].

A generic rank two distributions in dimension five is a rank two subbundle D in the tangent bundle of a 5-manifold M such that Lie brackets of sections of D span a subbundle [ D , D ] of rank three, and triple brackets of sections of D span all of the tangent bundle, [ D , [ D , D ] ] = T M . These geometric structures have first been studied by Cartan [16]. The Lie bracket of vector fields induces a fiberwise Lie bracket on the associated graded bundle,

This is a locally trivial bundle of graded Lie algebras over M called the bundle of osculating algebras. Its fibers are all isomorphic to the five-dimensional graded nilpotent Lie algebra

with graded basis X 1 , X 2 ∈ g − 1 , X 3 ∈ g − 2 , X 4 , X 5 ∈ g − 3 , and brackets

The simply connected Lie group with Lie algebra g will be denoted by G . The left invariant two-plane field spanned by X 1 and X 2 provides a basic example of a (2, 3, 5) distribution on G . Unlike contact or Engel structures, (2, 3, 5) distributions do have local geometry. A distribution of this type is locally diffeomorphic to the aforementioned left invariant distribution on G if and only if Cartan’s [16] harmonic curvature tensor, a section of S 4 D * , vanishes. This curvature tensor is constructed using an equivalent description of (2, 3, 5) distributions as regular normal parabolic geometries of type ( G 2 , P ) , where G 2 denotes the split real form of the exceptional Lie group and P denotes the parabolic subgroup corresponding to the longer root, see [16,56] or [15, Section 4.3.2]. Generic rank two distributions in dimension five have attracted quite some attention recently, cf. [2,3,8,9,12–14,21,28–32,38,43,55–58].

The Rumin complex associated with a (2, 3, 5) distribution on a five-manifold M is a natural sequence of higher order differential operators:

where ℋ q ( t M ) denotes the vector bundle obtained by passing to the fiberwise Lie algebra cohomology of t M with trivial coefficients. The Betti numbers are rk ℋ q ( t M ) = dim H q ( g ) = 1 , 2, 3, 2, 1 for q = 0 , … , 4 and the Heisenberg order of the Rumin differential D q is k q = 1 , 3, 2, 3, 1 for q = 0 , … , 4 , see [11, Section 5] and [22, Example 4.21]. The Rumin differentials form a complex, D q + 1 D q = 0 , that computes the de Rham cohomology of M . Actually, there exist injective differential operators L q : Γ ( ℋ q ( t M ) ) → Ω q ( M ) embedding the Rumin complex as a subcomplex in the de Rham complex and inducing isomorphisms on cohomology. Twisting with a flat complex vector bundle F , we obtain a complex of differential operators

computing H * ( M ; F ) , the de Rham cohomology of M with coefficients in F . Rumin has shown that the sequence (1.2) becomes exact on the level of the Heisenberg principal symbol, see [50, Theorem 3], [52, Theorem 5.2], or [22, Corollary 4.18]. Hence, the Rumin complex is a Rockland [47] complex, the analogue of an elliptic complex in the Heisenberg calculus; see [22, Section 2.3] for more details.

A fiberwise graded Euclidean inner product g on t M and a fiberwise Hermitian inner product h on F give rise to L 2 inner products on Γ ( ℋ q ( t M ) ⊗ F ) , which in turn provide formal adjoints D q * of the Rumin differentials in (2). Assuming M to be closed, the operator D q * D q has an infinite dimensional kernel if q > 0 , but the remaining part of its spectrum consists of isolated positive eigenvalues with finite multiplicities only. Moreover, ( D q * D q ) − s is trace class for ℜ s > 10 ⁄ 2 k q . The number ten appears here because this is the homogeneous dimension of the filtered manifold M . Furthermore, the function tr ( D q * D q ) − s admits an analytic continuation to a meromorphic function on the entire complex plane, which is holomorphic at s = 0 , see [29, Remark 2.9]. This permits to define the zeta regularized determinant det * ∣ D q ∣ by

The notation det * indicates that the zero eigenspace does not contribute, i.e., we are considering the regularized product of nonzero eigenvalues. Correspondingly, the complex powers are defined to vanish on the kernel of D q * D q . The analytic torsion τ ( M , D , F , g , h ) is the graded determinant of the Rumin complex, i.e.,

By Poincaré duality, see [50, Section 2], [52, Proposition 2.8], or [29, Section 3.3], we have det * ∣ D 0 ∣ = det * ∣ D 4 ∣ and det * ∣ D 1 ∣ = det * ∣ D 3 ∣ , provided h is parallel. Hence, in this (unitary) case, the torsion may be expressed in terms of three determinants,

Since g has pure cohomology, the grading automorphism ϕ τ ∈ Aut gr ( g ) acts as a scalar on each cohomology, namely, by τ N q on H q ( g ) , where N q = 0 , 1 , 4 , 6 , 9 , 10 for q = 0 , … , 5 . These numbers are related to the Heisenberg orders of the Rumin differentials via the equation k q = N q + 1 − N q . Let κ denote a common multiple of the differential orders k 0 , … , k 4 . Hence, there exist natural numbers a q such that κ = a q k q . The smallest possible choice would be the lowest common multiple κ = 6 with a q = 6 , 2 , 3 , 2 , 6 for q = 0 , … , 4 . Then the Rumin-Seshadri operators

are all of Heisenberg order 2 κ . These operators are analytically much better behaved than D q * D q . Indeed, Δ q is a Rockland [47] operator and, thus, admits a parametrix in the Heisenberg calculus [22,40,59]. Moreover, Δ q − s is trace class for ℜ s > 10 ⁄ 2 κ , and the zeta function tr Δ q − s admits an analytic continuation to a meromorphic function on the entire complex plane, which is holomorphic at s = 0 , see [23, Corollary 2]. The analytic properties of the operator D q * D q stated in the preceding paragraph can be readily deduced from the corresponding properties of Δ q , see [29, Remark 2.9] for more details. By

the analytic torsion of the Rumin complex can be expressed in the following form:

which is analogous to the formulas for the Ray-Singer torsion in [45, Definition 1.6] and the Rumin-Seshadri torsion in [54, p. 728]; see [29, Eq. (34)] for more details. The good behavior of the analytic torsion function in (1.6), for instance, with respect to variation of metric choices [29, Section 2.4], serves as motivation for using powers in (1.5) such that the operators Δ q all have the same Heisenberg order.

It turns out to be convenient to incorporate the zero eigenspaces of Δ q and consider the analytic torsion of the Rumin complex as a norm ‖ − ‖ D , g , h sdet H * ( M ; F ) on the graded determinant line

Basic properties of this torsion have been established in [29]. In the acyclic case, it reduces to (the reciprocal of) τ ( M , D , F , g , h ) . The Ray-Singer torsion, too, is best regarded as a norm on this graded determinant line [6,7]. We will denote it by ‖ − ‖ RS sdet H * ( M ; F ) . The Ray-Singer torsion does not depend on metric choices since the dimension of M is odd.

In this article we will determine the analytic torsion of the Rumin complex on nilmanifolds Γ \ G , where Γ is a lattice in G , i.e., a cocompact discrete subgroup in G . We consider any left invariant (2, 3, 5) distribution D G on G and any left invariant fiberwise graded Euclidean inner product on t G . These descend to a (2, 3, 5) distribution D Γ \ G on the nilmanifold Γ \ G and a fiberwise graded Euclidean inner product g Γ \ G on t ( Γ \ G ) . For a unitary representation ρ : Γ → U ( k ) , we let h ρ denote the canonical (parallel) fiberwise Hermitian inner product on the associated flat complex vector bundle F ρ ≔ G × ρ C k over Γ \ G .

We are now in a position to state our main result.

In the situation of this theorem, the Ray-Singer torsion is known to be trivial too, τ RS ( Γ \ G ; F χ ) = 1 , see Lemma 6.1. Hence, it coincides with the analytic torsion of the Rumin complex. More generally, we have:

These are the first (2, 3, 5) distributions for which the analytic torsion of the Rumin complex has been computed. In [29, Theorem 1.3], a partial result of this type has been obtained by exploiting a large discrete symmetry group of the distribution D Γ \ G .

The proof given below is based on the decomposition of the Rumin complex over Γ \ G into a countable direct sum D * = ⊕ ρ m ( ρ ) ⋅ ρ ( D * ) of Rumin complexes in irreducible unitary representations ρ of G , denoted by ρ ( D * ) . The multiplicities m ( ρ ) of the contributing representations are known explicitly through a formula due to Howe [33] and Richardson [46] and involve counting the number of solutions to certain quadratic congruences, cf. Lemma 3.2(III), (3.13), or (4.12). The zeta function in (1.6) decomposes accordingly,

where ζ ρ ( s ) denotes the zeta function associated with ρ ( D * ) . The values ζ ρ ( 0 ) and ζ ρ ′ ( 0 ) have been determined in [30] for every irreducible unitary representation ρ . Even though the sum on the right-hand side in (1.8) converges only for ℜ s > 10 ⁄ 2 κ , we are able to conclude ζ Γ \ G , D Γ \ G , F χ , g Γ \ G , h χ ′ ( 0 ) = 0 via regularization, whence the theorem stated earlier. This is the subtlest part of the article at hand and builds on further properties of ζ ρ ( s ) obtained in [30]. Decomposing

according to the three types of irreducible unitary representations of G , we will use Epstein zeta functions to obtain the analytic continuation of each of the three summands, cf. (5.3), (5.8), and (5.23).

Alternatively, one could try to extend Albin and Quan’s [1] analysis of the sub-Riemannian limit to show that the torsion of the Rumin complex of a (2, 3, 5) distribution differs from the Ray-Singer torsion by the integral of a local quantity cf. [1, Corollary 3]. This, too, would immediately imply the corollary stated earlier.

The remaining part of this article is organized as follows. In Section 2, we provide an explicit description of all lattices in G . In Section 3, we use the aforementioned result due to Howe [33] and Richardson [46] to decompose the space of sections of F χ into irreducible G -representations. In Section 4, we describe the corresponding decomposition of the zeta function ζ Γ \ G , D Γ \ G , F χ , g Γ \ G , h χ ( s ) . In Section 5, building on results obtained in [30], we evaluate the derivative of this zeta function at s = 0 . In Section 6, we derive the theorem and corollary stated earlier.

2 Lattices

In this section, we provide explicit descriptions of all lattices in G .

Let X 1 , … , X 5 be a graded basis of g with brackets as in (1.1). For the sake of notational simplicity, we will use this basis to identify g with R 5 , that is, a vector ( x 1 , … , x 5 ) t ∈ R 5 will be identified with ∑ i = 1 5 x i X i ∈ g .

The exponential map provides a diffeomorphism exp : g → G . By using the Baker-Campbell-Hausdorff formula, we find

where

The center of G will be denoted by Z . Clearly, Z = exp ( z ) , where z denotes the center of g , which is spanned by X 4 , X 5 . For the group of commutators, we have [ G , G ] = exp ( [ g , g ] ) and the derived subalgebra [ g , g ] is spanned by X 3 , X 4 , X 5 .

For r ∈ N and e , f , g , h , u , v ∈ Q , we consider γ ˜ i ∈ g defined by

and let Γ denote the subgroup of G generated by the exponentials:

Let Γ ″ denote the lattice in R 2 generated by

The following generalizes [29, Lemma 4.4].

The natural homomorphism p : G → G ⁄ [ G , G ] ≅ R 2 gives rise to a short exact sequence of abelian groups:

where Γ ∩ [ G , G ] [ Γ , Γ ] is a finite abelian group and p ( Γ ) ≅ Z 2 . In particular, the sequence splits, and we obtain an isomorphism

The group of unitary characters of Γ , thus, is a finite union of 2-tori,

where A = hom Γ ∩ [ G , G ] [ Γ , Γ ] , U ( 1 ) is a finite abelian group.

To specify a character χ : Γ → U ( 1 ) , it suffices to know its values on the generators, χ ( γ i ) for i = 1 , … , 5 .

3 Decomposition into irreducible representations

If Γ is a lattice in a simply connected nilpotent Lie group N , then L 2 ( Γ \ N ) decomposes into a countable direct sum of irreducible unitary representations of N . The multiplicities of the representations appearing in this decomposition have been studied by Moore [41]. An explicit formula for these multiplicities has been conjectured by Mostow and proved, independently, by Howe [33] and Richardson [46]. More generally, for every unitary character χ : Γ → U ( 1 ) , the induced representation

decomposes into a countable direct sum of irreducible unitary representations and the multiplicities are known explicitly, cf. [33, Theorem 1] and [46, Theorem 5.3].

Let us briefly recall some of the terminology used in the preceding lemma. To this end, let n denote the Lie algebra of N , suppose f ∈ n * , and let m denote a maximal subordinate subalgebra, i.e., a subalgebra of maximal dimension in n such that f ( [ m , m ] ) = 0 . Then

defines a unitary character on the group M ≔ exp ( m ) . According to Kirillov [19,34,35], such a character induces an irreducible unitary representation of N given by right translation on the Hilbert space

One may always assume m to be special [46, §3] and then ( f ¯ , M ) is called a maximal character [46, §4]. The group N acts by conjugation on the set of maximal characters [46, Lemma 4.1] and the stabilizer of ( f ¯ , M ) is M , see [46, Lemma 5.1]. A maximal character ( f ¯ , M ) is called rational [46, §3] if it can be obtained from a (rational) linear functional f ∈ n * mapping log Γ into the rational numbers and m is rational with respect to the rational structure on n provided by log Γ . Note that m is rational if and only if Γ ∩ M is a lattice in M , i.e., if and only if ( Γ ∩ M ) \ M is compact. Clearly, the action of Γ preserves the subset of rational maximal characters and the integrality condition f ¯ ∣ Γ ∩ M = χ ∣ Γ ∩ M .

In the remaining part of this section, we will specialize the preceding lemma to the five-dimensional Lie group G considered earlier. All irreducible unitary representations of this Lie group have been described explicitly by Dixmier in [24, Proposition 8]. There are three types of irreducible unitary representations of G , which we now describe in terms of a graded basis X 1 , … , X 5 of g satisfying (1.1). If ρ is a representation of G , we let ρ ′ denote the corresponding infinitesimal representation of g .